# Why does energy flow downhill/entropy increase?

I'm not sure if this is asked anywhere else before, but so far my search isn't fruitful.

I'm in my second bachelor of physics and mathematics, and currently taking a thermodynamics course. We just got introduced to the second law and entropy, but I am confused by its deeper meaning. I understand that entropy always increases and in our course text (An introduction to thermal physics by Daniel V. Schroeder) this is explained by microstates of a system. I understand that systems with more 'disorder' have a higher chance of occuring than systems with a lower disorder. The thing that bothers me right now is why nature obeys this statistical argument. Just because combinatorics says a certain configuration is more likely doesn't imply that nature will develop to that situation.

My question is: why does a system tend to a configuration with more disorder.

I've read on several pages that this is because 'energy flows downhill'. I can see how this solves my question, because if the energy flows downhill, then it's quite easy to see that the result is a system with energy more spread out, a system with more disorder. But then again why does energy flow downhill? What drives the fluid of energy between systems? On some internet pages, the answer to this question is 'because entropy increases', but this brings us back to the original problem.

In conclusion: why does energy flow downhill or why does entropy increase, explained on a fundamental level.

• I've read on several pages that this is because 'energy flows downhill Eh kind of. Although there are processes where entropy increases without energy changing. – Aaron Stevens Mar 15 at 13:14
• Also, it seems like you are questioning the assumption that all microstates are equally likely, right? Because it's that assumption that then leads to everything else. Your statement Just because combinatorics says a certain configuration is more likely doesn't imply that nature will develop to that situation. Seems odd because that is precisely what "more likely" means in terms of the extremely large numbers we deal with. – Aaron Stevens Mar 15 at 13:15
• Thank you for your answer @AaronStevens. Isn't the assumption different than what you said: namely that some microstates are more likely than others? – Thibeau Wouters Mar 15 at 13:18
• Not at all. If you are using Schroeder I would suggest looking at the text again. He discusses this. The assumption is that all microstates are equally likely, which makes certain macrostates with more microstates associated with them much more likely. – Aaron Stevens Mar 15 at 13:22
• @ThibeauWouters it doesn't do that. The system can even change its microstate arriving to ordered configuration, that's not forbidden, it's just not likely. Max. Entropy doesn't mean that the system goes always to the most disordered microstate. Entropy is roughly speaking, the number of the accessible microstates for the given system, so the bigger this number is and the bigger the entropy is – Run like hell Mar 15 at 14:11

You say

My question is: why does a system tend to a configuration with more disorder.

Try to think it in this way:

1. There is a macrostate of fixed energy of a system and you can measure properties of this macrostate
2. There are multiple microstates corresponding to this macrostate, each of this microstates is equally probable
3. The "ordered" states are very few compared to the "disordered" ones
4. Since the probability that a microstate is realized is equal to the probability that another microstate is realized, from the point $$3$$ it follows that, because the disordered configurations are way way more than the ordered ones, it's way more likely that the system is in one of these disordered configurations rather than in an ordered one.

As far as I got your question your problem is with the point $$2$$. Why is it valid? It's an hypothesis, called hypothesis of equal a priori probability. So why should we trust this hypothesis? Because it works, that is after you do such hypothesis you study the consequences of this hypothesis and then compare them with the experiment. If the experiments are consistent with it, it means that your hypothesis was a good one.

Addendum after the comments: Notice that the system can even change its microstate arriving to ordered configurations, that's not forbidden, it's just not likely. Maximum Entropy doesn't mean that the system always ends up in the most disordered microstate. Entropy is, roughly speaking, the number of the accessible microstates for the given system, so the bigger this number is and the bigger the entropy is

• it follows that the disordered microstates are largely favoured. Do you mean macrostate? One microstate isn't more favored over another. – Aaron Stevens Mar 15 at 13:54
• @AaronStevens No, with macrostate I mean a system described for example by its temperature and pressure, with microstate I mean all possible different configurations that macroscopically give rise to this observable macrostate. Some configurations are ordered, others aren't. – Run like hell Mar 15 at 13:58
• @AaronStevens I should probably rephrase it. I mean that since there is a huge number of disordered microstates compared to the ordered ones, it's more likely that the system is in one of these disordered microstates – Run like hell Mar 15 at 13:59
• Ok yes I agree. It was just the wording – Aaron Stevens Mar 15 at 14:04
• @AaronStevens Regarding the answer, I understand the point you made, and I believe in the hypothesis because experiments confirm this. Yet as I said to Aaron Stevens in the comment section below the question, my main difficulty is the change leading to the macrostate with the most disordered configuration. – Thibeau Wouters Mar 15 at 14:05

I'll give a hypothetical example in which entropy decreases.

I assume you have some familiarity with phase space - plots of the position versus momentum of a system. If not, this Wikipedia article has good illustrations.

Suppose the phase space for some system has an attractor, meaning all the paths tend to go to one spot.

Public Domain, https://commons.wikimedia.org/w/index.php?curid=556459

Physically, you could think of a harmonic oscillator with some damping. No matter the initial trajectory of the oscillator, it always settles down to its equilibrium position.

In a universe where the fundamental laws physics behave like that in phase space, entropy could decrease and the second law would be false. The reasoning is simple. Suppose the macrostate is something like "the energy of the oscillator is between $$E$$ and $$E + \mathrm{d}E$$". The possible states (i.e. "number of microstates") of the oscillator would be a thin circular ring on the phase space diagram, and the entropy would be a function of the area of that ring, called the "phase space volume".

Then as time goes on, the energy of the oscillator would go down and the ring would get smaller, eventually becoming just a dot. The phase space volume would decrease. Then the number of microstates corresponding to the macrostate would have gone down, and the entropy would have decreased.

The reason this doesn't apply to our universe is that the fundamental laws of physics have a special form. In fact, at the fundamental level, there is no such thing as damping. Of course, in real life, we can build a damped oscillator. But in order to do that, the oscillator needs to be interacting with something else; maybe there's friction from some rubbing. And that friction heats up the molecules in whatever the oscillator is rubbing against, increasing the energy and the phase space volume over in that system. So whenever we decrease the phase space volume in one place, we wind up increasing it even more somewhere else.

The laws of physics actually say that if you have an isolated system, the phase space volume is constant. This is Liouville's Theorem, or in quantum mechanics, unitarity.

In a universe where this were not true, entropy could decrease.

This might not be the entire story of why entropy increases, but no explanation of why entropy increases that fails to mention this property of physical law can possibly be complete.

• This is a great and quite different approach to the question, I really like it. I know Liouville's theorem, but somehow I haven't seen the link with entropy until now. I'll surely look into this later on with more focus. Thanks a lot for sharing! – Thibeau Wouters Mar 15 at 14:39

As stated in previously posted answers: the reasoning is probabilistic.

I remember reading about an extremely simplified model, I think by Peter Atkins.

Imagine an array of playing cards 10 by 5 cards. The initial state is that 25 cards are open, arranged 5 by 5, and 25 cards are closed, also arranged 5 by 5.

Given that initial state, start swapping pairs of adjacent cards randomly. Over time the separated state will evolve to a state where closed and open cards are present in the entire 10 by 5 array. The driving effect is that the pairs are swapped randomly.

Avoid the downhill metaphor

Actually, it's important to think of the move towards largest probability as distinct from moving down an energy gradient.

An example of 'moving downhill' would be geologic erosion. Over geologic timescales mountains erode. Unless there is some plate tectonics moving things up any landscape features will tend to flatten over time.

However, and this is crucial to be aware of, there are cases where an increase of entropy can cause movement against an energy gradient.

One example of that is osmotic pressure

Let's take the mose usual case: water as the solvent. You have a membrane that is permeable to water molecules, but not permeable to larger molecules that are dissolved in the water.

At the membrane/water interface water molecules are entering and leaving the membrane. When there are large molecules present in the water the probability of water molecules to enter the membrane is decreased (compared to the case of pure water), the large molecules are getting in the way. But the probability of water molecules to leave the membrane is not decreased. As a consequence, there will be a net flow of water across the membrane, building up a pressure differential.

The expression 'energy flows downhill' is intended to capture a probabilistic effect. If you drop an object from some height then just prior to impact the object has significant kinetic energy, which is low entropy energy. On impact that kinetic energy becomes heat. Heat is a form of kinetic energy too (velocity of the atoms/molecules that the object is comprised of), but heat is a higher entropy form of energy.

Energy is always conserved, but probability constrains it to only move in the direction of increasing entropy.

All things want to have the lowest potential energy they can achieve . That is why when leave a ball from some height the ball starts falling ( it's gravitational potential is reduced ) .Entropy is how messy things are . When the universe began , the entropy was very small because all of the energy of the universe was in one particular point ( the singularity) . Now we have lot of entropy because things are messy . For example , when you go to a bar everyone is sitting and drinking so they entropy is low . But if a warmhead enters the bar and starts assaulting poeple , then the customers will defend themselvels , things will be broken and the bar will be more messy.

• This doesn't answer the question – Aaron Stevens Mar 15 at 13:18
• What do you mean it doesnt answer the question? – Altair Mar 15 at 13:20
• Your answer essentially says that entropy increases, which the OP knows already. – Aaron Stevens Mar 15 at 13:24